How to estimate the point of divergence between two continuous time survival curves? In this experiment we collect $N$ samples and each sample yields a pair of survival curves. The two survival curves are hypothesized to be identical up until time $t$ and diverge thereafter. What would be an appropriate method by which such a point $t$ can be estimated and statistically validated?
 A: One method that might be useful is change point detection.
https://en.wikipedia.org/wiki/Change_detection
There are many variants, but in general you partition the data into two sets and compute a test statistic, such as the Kolmogov-Smirnov, to determine whether the two distributions are the same.  If null hypothesis is rejected, there is a break or "change point" at time t.  In order to determine all such times, the test is repeated for each time step.
A: There is some ambiguity in what the OP means by the "pair of survival curves" obtained from each of the "N samples."
If the "survival curves" are changes in the value of some continuous variable as a function of time from a maximum initial value down to zero for two cases or two groups within each of the N samples, then any of a variety of change-point approaches could work in principle, as suggested in another answer. The task would be simplified if there were an underlying theoretical model, such as exponential decay with a time constant that changes for one of the pair during the period of observation.
If the "survival curves" are time-to-event data for two groups distinguished within each of the N samples, then the problem is more complicated. A survey of change-point packages in R found only one that accepts survival models, with an approach that is "not guaranteed to work," according to its manual, for model types other than lm or glm. Even that package does not meet the requirements of the OP, as it looks for change-points for regression coefficients within values of a continuous predictor variable, not for change points in time as the OP seeks.
So for time-to-event data what's needed is a way to search for the time of a step change in the relationship between the group membership within each sample and survival. For a Cox survival model that would be the time of a step change in the regression coefficient for group, where group represents which pair within a sample the data are from. The flexible survival regression approaches provided by the timereg package seem to be designed for coefficients that are continuous functions of time rather than step functions, but might serve this purpose. Alternatively, there might be ways to adapt the approach to step-function coefficient changes described in the time-dependent vignette for the R survival package. As that splits data into strata based on the suspected time of the step change there would have to be some iterative approach to estimate the actual divergence time, and I'm not sure what would need to be done to estimate confidence intervals or to avoid overfitting with that approach.
Finally, for distinguishing when Kaplan-Meier survival curves diverge, the bpcp package uses the individual KM curves to estimate their differences and associated CI at fixed times. Applied over a set of times, that approach could be used to determine estimated survival differences over time. Change-point analysis might then identify a time of divergence from 0 difference. Bootstrapping could presumably "statistically validate" the results.
